On c-completely regular frames

Document Type : Original Article


1 Esfarayen University of Technology, Esfarayen, North Khorasan, Iran.

2 Ali Akbar Estaji, Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.


Motivated by definitions of countable completely regular spaces and completely below relations of frames, we define what we call a $c$-completely below relation, denoted by $\prec\!\!\prec_c$, in between two elements of a frame. We show that $a\prec\!\!\prec_c b$ for two elements $a, b$ of a frame $L$ if and only if there is $\alpha\in\mathcal{R}L$ such that $\coz\alpha\wedge a=0$ and $\coz(\alpha-{\bf1})\leq b$ where the set $\{r\in\mathbb{R} : \coz(\alpha-{\bf r})\ne 1\}$ is countable. We say a frame $L$ is a $c$-completely regular frame if $a=\bigvee \limits_{x\prec\!\!\prec_ca}x$ for any $a\in L$. It is shown that a frame $L$ is a $c$-completely regular frame if and only if it is a zero-dimensional frame. An ideal $I$ of a frame $L$ is said to be $c$-completely regular if $a\in I$ implies $a\prec\!\!\prec_c b$ for some $b\in I$. The set of all $c$-completely regular ideals of a frame $L$, denoted by ${\mathrm{c-CRegId}}(L)$, is a compact regular frame and it is a compactification for $L$ whenever $it$ is a $c$-completely regular frame. We denote this compactification by $\beta_cL$ and it is isomorphic to the frame $\beta_0L$, that is, Stone-Banaschewski compactification of $L$. Finally, we show that open and closed quotients of a $c$-completely regular frame are $c$-completely regular.


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